The Definition of Differential

This section will be expanded to the field of multivariate function in the future.

But for now, we talk about simple functions and curves to indicate the definition of differential.

From Derivative to Differential

For a function defined in a region, when a small change taken in ${\displaystyle x}$ noted by ${\displaystyle dx}$, we have a corresponded change in ${\displaystyle y}$ which can be expressed as

${\displaystyle dy=f(x+dx)-f(x)----e.1}$

Since we are talking about small changes, we just focus on ${\displaystyle dx}$ and ${\displaystyle dy}$.
In equation ${\displaystyle e.1}$, we can transform it into another form like this:

${\displaystyle dy=Adx+o(dx)}$

And from the section of infinitesimals we know that ${\displaystyle o(dx)}$ disappears more and more faster than ${\displaystyle dx}$, so we ignore it.
And it becomes:

${\displaystyle dy=Adx}$

Fortunately, we find that ${\displaystyle A}$ in this equation is the derivative of ${\displaystyle y=f(x)}$. Actually, ${\displaystyle dy}$ is the linear approaches that shows the real change of the function at some point by taking a little step in ${\displaystyle x}$ noted as ${\displaystyle dx}$.

For simple curves like a circle that could be departed as several functions, the definition above also works. But when going deeper to the field of multivariate functions, we need a stronger tool to handle with their derivatives.

But for now, we just need to understand the simple definition of differential. It will help a lot in the section of derivatives and integrals.